2013 Mock AIME I Problems/Problem 6
Problem 6
Find the number of integer values can have such that the equation
has a solution.
Solution
is a continuous function, so every value between its minimum and maximum is attainable. By Cauchy-Schwarz,
Giving a maximum of
, which is achievable when
. Note that a minimum of
can be attained at
. Thus the values of
that work are the integers from
to
, inclusive, giving a total of
.